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Difference between revisions of "2021 AMC 10B Problems/Problem 21"

(→‎Video Solution by OmegaLearn (Using Pythagoras Theorem and Similar Triangles))
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A square piece of paper has side length <math>1</math> and vertices <math>A,B,C,</math> and <math>D</math> in that order. As shown in the figure, the paper is folded so that vertex <math>C</math> meets edge <math>\overline{AD}</math> at point <math>C'</math>, and edge <math>\overline{BC}</math> intersects edge <math>\overline{AB}</math> at point <math>E</math>. Suppose that <math>C'D = \frac{1}{3}</math>. What is the perimeter of triangle <math>\bigtriangleup AEC' ?</math>
 
A square piece of paper has side length <math>1</math> and vertices <math>A,B,C,</math> and <math>D</math> in that order. As shown in the figure, the paper is folded so that vertex <math>C</math> meets edge <math>\overline{AD}</math> at point <math>C'</math>, and edge <math>\overline{BC}</math> intersects edge <math>\overline{AB}</math> at point <math>E</math>. Suppose that <math>C'D = \frac{1}{3}</math>. What is the perimeter of triangle <math>\bigtriangleup AEC' ?</math>
  
−
<math>\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\sqrt{136} \qquad\textbf{(D)} ~1 + \frac{3}{4}\sqrt{3} \qquad\textbf{(E)} ~\frac{7}{3}</math>
+
<math>\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\frac{13}{6} \qquad\textbf{(D)} ~1 + \frac{3}{4}\sqrt{3} \qquad\textbf{(E)} ~\frac{7}{3}</math>
 
<asy>
 
<asy>
 
/* Made by samrocksnature */
 
/* Made by samrocksnature */
Line 8: Line 8:
 
pair CC=(0.666666666666,1);
 
pair CC=(0.666666666666,1);
 
pair D=(1,1);
 
pair D=(1,1);
−
pair F=(1,0.62);
+
pair F=(1,0.440062);
 
pair C=(1,0);
 
pair C=(1,0);
 
pair B=(0,0);
 
pair B=(0,0);
−
pair G=(0,0.25);
+
pair G=(0,0.22005);
 
pair H=(-0.13,0.41);
 
pair H=(-0.13,0.41);
 
pair E=(0,0.5);
 
pair E=(0,0.5);
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label("E",E,NW);
 
label("E",E,NW);
 
label("C'",CC,N);
 
label("C'",CC,N);
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label("F",F,NE);
 
</asy>
 
</asy>
  
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<cmath>x^2 + \left(\frac{1}{3}\right)^2 = (1-x)^2 \rightarrow x^2 + \frac{1}{9} = x^2 - 2x + 1 \rightarrow x=\frac{4}{9}</cmath>
 
<cmath>x^2 + \left(\frac{1}{3}\right)^2 = (1-x)^2 \rightarrow x^2 + \frac{1}{9} = x^2 - 2x + 1 \rightarrow x=\frac{4}{9}</cmath>
  
−
We know this is a 3-4-5 triangle because the side lengths are <math>\frac{3}{9}, \frac{4}{9}, \frac{5}{9}</math>. We also know that <math>EAC'</math> is similar to <math>C'DF</math> because angle <math>C'</math> is a right angle. Now, we can use similarity to find out that the perimeter is just the perimeter of <math>C'DF * \frac{AC'}{DF}</math>. Thats just <math>\frac{4}{3} * \frac{\frac{2}{3}}{\frac{4}{9}} = \frac{4}{3} * \frac{3}{2} = 2</math>. Therefore, the final answer is <math>\boxed{A}</math>
+
We know this is a 3-4-5 triangle because the side lengths are <math>\frac{3}{9}, \frac{4}{9}, \frac{5}{9}</math>. We also know that <math>EAC'</math> is similar to <math>C'DF</math> because angle <math>EC'F</math> is a right angle. Now, we can use similarity to find out that the perimeter is just the perimeter of <math>C'DF \times \frac{AC'}{DF}</math>. That's just <math>\frac{4}{3} \times \frac{\frac{2}{3}}{\frac{4}{9}} = \frac{4}{3} \times \frac{3}{2} = 2</math>. Therefore, the final answer is <math>\boxed{\textbf{(A)} ~2}</math>
  
 
~Tony_Li2007
 
~Tony_Li2007
  
 
==Solution 2==
 
==Solution 2==
−
Let line we're reflecting over be <math>\ell</math>, and let the points where it hits <math>AB</math> and <math>CD</math>, be <math>M</math> and <math>N</math>, respectively. Notice, to reflect over a line we find the perpendicular passing through the midpoint of the two points (which are the reflected and the original). So, we first find the equation of the line <math>\ell</math>. The segment <math>CC'</math> has slope <math>\frac{0 - 1}{1 - 2/3} = -3</math>, implying line <math>\ell</math> has a slope of <math>\frac{1}{3}</math>. Also, the midpoint of segment <math>CC'</math> is <math>\left( \frac{5}{6}, \frac{1}{2} \right)</math>, so line <math>\ell</math> passes through this point. Then, we get the equation of line <math>\ell</math> is simply <math>y = \frac{1}{3} x + \frac{2}{9}</math>. Then, if the point where <math>B</math> is reflected over line <math>\ell</math> is <math>B'</math>, then we get <math>BB'</math> is the line <math>y = -3x</math>. The intersection of <math>\ell</math> and segment <math>BB'</math> is <math>\left( - \frac{1}{15}, \frac{1}{5} \right)</math>. So, we get <math>B' = \left(- \frac{2}{15}, \frac{2}{5} \right)</math>. Then, line segment <math>B'C'</math> has equation <math>y = \frac{3}{4} x + \frac{1}{2}</math>, so the point <math>E</math> is the <math>y</math>-intercept, or <math>\left(0, \frac{1}{2} \right)</math>. This implies that <math>AE = \frac{1}{2}, AC' = \frac{2}{3}</math>, and by the Pythagorean Theorem, <math>EC' = \frac{5}{6}</math> (or you could notice <math>\triangle AEC'</math> is a <math>3-4-5</math> right triangle). Then, the perimeter is <math>\frac{1}{2} + \frac{2}{3} + \frac{5}{6} = 2</math>, so our answer is <math>\boxed{\textbf{(A)} ~2}</math>. ~rocketsri
+
Let the line we're reflecting over be <math>\ell</math>, and let the points where it hits <math>AB</math> and <math>CD</math>, be <math>M</math> and <math>N</math>, respectively. Notice, to reflect over a line we find the perpendicular passing through the midpoint of the two points (which are the reflected and the original). So, we first find the equation of the line <math>\ell</math>. The segment <math>CC'</math> has slope <math>\frac{0 - 1}{1 - 2/3} = -3</math>, implying line <math>\ell</math> has a slope of <math>\frac{1}{3}</math>. Also, the midpoint of segment <math>CC'</math> is <math>\left( \frac{5}{6}, \frac{1}{2} \right)</math>, so line <math>\ell</math> passes through this point. Then, we get the equation of line <math>\ell</math> is simply <math>y = \frac{1}{3} x + \frac{2}{9}</math>. Then, if the point where <math>B</math> is reflected over line <math>\ell</math> is <math>B'</math>, then we get <math>BB'</math> is the line <math>y = -3x</math>. The intersection of <math>\ell</math> and segment <math>BB'</math> is <math>\left( - \frac{1}{15}, \frac{1}{5} \right)</math>. So, we get <math>B' = \left(- \frac{2}{15}, \frac{2}{5} \right)</math>. Then, line segment <math>B'C'</math> has equation <math>y = \frac{3}{4} x + \frac{1}{2}</math>, so the point <math>E</math> is the <math>y</math>-intercept, or <math>\left(0, \frac{1}{2} \right)</math>. This implies that <math>AE = \frac{1}{2}, AC' = \frac{2}{3}</math>, and by the Pythagorean Theorem, <math>EC' = \frac{5}{6}</math> (or you could notice <math>\triangle AEC'</math> is a <math>3-4-5</math> right triangle). Then, the perimeter is <math>\frac{1}{2} + \frac{2}{3} + \frac{5}{6} = 2</math>, so our answer is <math>\boxed{\textbf{(A)} ~2}</math>. ~rocketsri
  
 
==Solution 3 (Fakesolve):==
 
==Solution 3 (Fakesolve):==
−
Assume that E is the midpoint of <math>\overline{AB}</math>. Then, <math>\overline{AE}=\frac{1}{2}</math> and since <math>C'D=\frac{1}{3}</math>, <math>\overline{AC'}=\frac{2}{3}</math>. By the Pythagorean Theorem, <math>\overline{EC'}=\frac{5}{6}</math>. It easily follows that our desired perimeter is <math>2 \rightarrow \boxed{A}</math> ~samrocksnature
+
Assume that E is the midpoint of <math>\overline{AB}</math>. Then, <math>\overline{AE}=\frac{1}{2}</math> and since <math>C'D=\frac{1}{3}</math>, <math>\overline{AC'}=\frac{2}{3}</math>. By the Pythagorean Theorem, <math>\overline{EC'}=\frac{5}{6}</math>. It easily follows that our desired perimeter is <math>\boxed{\textbf{(A)} ~2}</math> ~samrocksnature
  
 
==Solution 4==
 
==Solution 4==
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−
Then, we can find we can find the length of <math>\overline{AE}</math> by expressing the length of <math>\overline{EF}</math> in two different ways, in terms of <math>AE</math>. If let <math>AE = a</math>, by the Pythagorean Theorem we have that <math>EC = \sqrt{a^2 + \left(\frac{2}{3}\right)^2} = \sqrt{a^2 + \frac{4}{9}}.</math> Therefore, since we know that <math>\angle EC'F</math> is right, by Pythagoras again we have that <math>EF = \sqrt{\left(\sqrt{a^2+\frac{4}{9}}\right)^2 + \left(\frac{5}{9}\right)^2} = \sqrt{a^2 + \frac{61}{81}}.</math>
+
Then, we can find we can find the length of <math>\overline{AE}</math> by expressing the length of <math>\overline{EF}</math> in two different ways, in terms of <math>AE</math>. If let <math>AE = a</math>, by the Pythagorean Theorem we have that <math>EC' = \sqrt{a^2 + \left(\frac{2}{3}\right)^2} = \sqrt{a^2 + \frac{4}{9}}.</math> Therefore, since we know that <math>\angle EC'F</math> is right, by Pythagoras again we have that <math>EF = \sqrt{\left(\sqrt{a^2+\frac{4}{9}}\right)^2 + \left(\frac{5}{9}\right)^2} = \sqrt{a^2 + \frac{61}{81}}.</math>
  
  
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Finally, we see that the perimeter of <math>\triangle AEC'</math> is <math>\frac{1}{2} + \frac{2}{3} + \sqrt{\left(\frac{1}{2}\right)^2 + \frac{4}{9}},</math> which we can simplify to be <math>2</math>. Thus, the answer is <math>\boxed{\textbf{(A)} ~2}.</math> ~laffytaffy
 
Finally, we see that the perimeter of <math>\triangle AEC'</math> is <math>\frac{1}{2} + \frac{2}{3} + \sqrt{\left(\frac{1}{2}\right)^2 + \frac{4}{9}},</math> which we can simplify to be <math>2</math>. Thus, the answer is <math>\boxed{\textbf{(A)} ~2}.</math> ~laffytaffy
 +
 +
==Solution 5 (Trig)==
 +
Draw a perpendicular line from <math>\overline{AB}</math> at <math>E</math>, and let it intersect <math>\overline{DC}</math> at <math>E'</math>. The angle between <math>\overline{AB}</math> and <math>\overline{EE'}</math> is <math>2\theta</math>, where <math>\theta</math> is the angle between the fold and a line perpendicular to <math>\overline{AD}</math>. The slope of the fold is <math>\frac{1}{3}</math> because it is perpendicular to <math>\overline{CC'}</math> (<math>\overline{CC'}</math> has a slope of <math>-3</math> using points <math>C'</math> and <math>C</math>, and perpendicular lines have slopes negative inverses of each other). Using tangent double angle formula, the slope of <math>\overline{EC'}</math> is <math>\frac{3}{4}</math>, which implies <math>\overline{AE}</math> = <math>\frac{1}{2}</math>. By the Pythagorean Theorem, <math>\overline{EC'}=\frac{5}{6}</math>. It easily follows that our desired perimeter is <math>\boxed{\textbf{(A)} ~2}</math> ~forrestc
 +
==Solution 6==
 +
It is easy to prove that the ratio of the sum of the larger leg and hypotenuse to the smaller leg depends monotonically on the angle of a right triangle, which means:
 +
<cmath>C' F + DF = CF + DF = CD = AD = 3 C'D \implies C' D : DF : C' F = 3 : 4 : 5.</cmath>
 +
 +
For a similar triangle, the ratio of the perimeter to the larger leg is <math>\frac {3 + 4 +5}{4} = 3.</math>
 +
 +
<math>\triangle AEC'  \sim \triangle DC'F \implies</math> the perimeter of triangle <math>\bigtriangleup AEC'</math> is <math>3 AC' = \boxed{\textbf{(A)} ~ 2}.</math>
 +
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
  
 
== Video Solution by OmegaLearn (Using Pythagoras Theorem and Similar Triangles) ==
 
== Video Solution by OmegaLearn (Using Pythagoras Theorem and Similar Triangles) ==
Line 65: Line 78:
  
 
~Interstigation
 
~Interstigation
 +
==Video Solution by The Power of Logic==
 +
https://www.youtube.com/watch?v=5kbQHcx1FfE
 +
 +
~The Power of Logic
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2021|ab=B|num-b=20|num-a=22}}
 
{{AMC10 box|year=2021|ab=B|num-b=20|num-a=22}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 23:15, 24 October 2024

Problem

A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Suppose that $C'D = \frac{1}{3}$. What is the perimeter of triangle $\bigtriangleup AEC' ?$

$\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\frac{13}{6} \qquad\textbf{(D)} ~1 + \frac{3}{4}\sqrt{3} \qquad\textbf{(E)} ~\frac{7}{3}$ [asy] /* Made by samrocksnature */ pair A=(0,1); pair CC=(0.666666666666,1); pair D=(1,1); pair F=(1,0.440062); pair C=(1,0); pair B=(0,0); pair G=(0,0.22005); pair H=(-0.13,0.41); pair E=(0,0.5); dot(A^^CC^^D^^C^^B^^E); draw(E--A--D--F); draw(G--B--C--F, dashed); fill(E--CC--F--G--H--E--CC--cycle, gray); draw(E--CC--F--G--H--E--CC); label("A",A,NW); label("B",B,SW); label("C",C,SE); label("D",D,NE); label("E",E,NW); label("C'",CC,N); label("F",F,NE); [/asy]

Solution 1

We can set the point on $CD$ where the fold occurs as point $F$. Then, we can set $FD$ as $x$, and $CF$ as $1-x$ because of symmetry due to the fold. It can be recognized that this is a right triangle, and solving for $x$, we get,

\[x^2 + \left(\frac{1}{3}\right)^2 = (1-x)^2 \rightarrow x^2 + \frac{1}{9} = x^2 - 2x + 1 \rightarrow x=\frac{4}{9}\]

We know this is a 3-4-5 triangle because the side lengths are $\frac{3}{9}, \frac{4}{9}, \frac{5}{9}$. We also know that $EAC'$ is similar to $C'DF$ because angle $EC'F$ is a right angle. Now, we can use similarity to find out that the perimeter is just the perimeter of $C'DF \times \frac{AC'}{DF}$. That's just $\frac{4}{3} \times \frac{\frac{2}{3}}{\frac{4}{9}} = \frac{4}{3} \times \frac{3}{2} = 2$. Therefore, the final answer is $\boxed{\textbf{(A)} ~2}$

~Tony_Li2007

Solution 2

Let the line we're reflecting over be $\ell$, and let the points where it hits $AB$ and $CD$, be $M$ and $N$, respectively. Notice, to reflect over a line we find the perpendicular passing through the midpoint of the two points (which are the reflected and the original). So, we first find the equation of the line $\ell$. The segment $CC'$ has slope $\frac{0 - 1}{1 - 2/3} = -3$, implying line $\ell$ has a slope of $\frac{1}{3}$. Also, the midpoint of segment $CC'$ is $\left( \frac{5}{6}, \frac{1}{2} \right)$, so line $\ell$ passes through this point. Then, we get the equation of line $\ell$ is simply $y = \frac{1}{3} x + \frac{2}{9}$. Then, if the point where $B$ is reflected over line $\ell$ is $B'$, then we get $BB'$ is the line $y = -3x$. The intersection of $\ell$ and segment $BB'$ is $\left( - \frac{1}{15}, \frac{1}{5} \right)$. So, we get $B' = \left(- \frac{2}{15}, \frac{2}{5} \right)$. Then, line segment $B'C'$ has equation $y = \frac{3}{4} x + \frac{1}{2}$, so the point $E$ is the $y$-intercept, or $\left(0, \frac{1}{2} \right)$. This implies that $AE = \frac{1}{2}, AC' = \frac{2}{3}$, and by the Pythagorean Theorem, $EC' = \frac{5}{6}$ (or you could notice $\triangle AEC'$ is a $3-4-5$ right triangle). Then, the perimeter is $\frac{1}{2} + \frac{2}{3} + \frac{5}{6} = 2$, so our answer is $\boxed{\textbf{(A)} ~2}$. ~rocketsri

Solution 3 (Fakesolve):

Assume that E is the midpoint of $\overline{AB}$. Then, $\overline{AE}=\frac{1}{2}$ and since $C'D=\frac{1}{3}$, $\overline{AC'}=\frac{2}{3}$. By the Pythagorean Theorem, $\overline{EC'}=\frac{5}{6}$. It easily follows that our desired perimeter is $\boxed{\textbf{(A)} ~2}$ ~samrocksnature

Solution 4

As described in Solution 1, we can find that $DF=\frac{4}{9}$, and $C'F = \frac{5}{9}.$


Then, we can find we can find the length of $\overline{AE}$ by expressing the length of $\overline{EF}$ in two different ways, in terms of $AE$. If let $AE = a$, by the Pythagorean Theorem we have that $EC' = \sqrt{a^2 + \left(\frac{2}{3}\right)^2} = \sqrt{a^2 + \frac{4}{9}}.$ Therefore, since we know that $\angle EC'F$ is right, by Pythagoras again we have that $EF = \sqrt{\left(\sqrt{a^2+\frac{4}{9}}\right)^2 + \left(\frac{5}{9}\right)^2} = \sqrt{a^2 + \frac{61}{81}}.$


Another way we can express $EF$ is by using Pythagoras on $\triangle XEF$, where $X$ is the foot of the perpendicular from $F$ to $\overline{AE}.$ We see that $ADFX$ is a rectangle, so we know that $AD = 1 = FX$. Secondly, since $FD = \frac{4}{9}, EX = a - \frac{4}{9}$. Therefore, through the Pythagorean Theorem, we find that $EF = \sqrt{\left(a-\frac{4}{9}\right)^2 + 1^2} = \sqrt{a^2 - \frac{8}{9}a +\frac{97}{81}}.$

Since we have found two expressions for the same length, we have the equation $\sqrt{a^2 + \frac{61}{81}} = \sqrt{a^2 - \frac{8}{9}a +\frac{97}{81}}.$ Solving this, we find that $a=\frac{1}{2}$.

Finally, we see that the perimeter of $\triangle AEC'$ is $\frac{1}{2} + \frac{2}{3} + \sqrt{\left(\frac{1}{2}\right)^2 + \frac{4}{9}},$ which we can simplify to be $2$. Thus, the answer is $\boxed{\textbf{(A)} ~2}.$ ~laffytaffy

Solution 5 (Trig)

Draw a perpendicular line from $\overline{AB}$ at $E$, and let it intersect $\overline{DC}$ at $E'$. The angle between $\overline{AB}$ and $\overline{EE'}$ is $2\theta$, where $\theta$ is the angle between the fold and a line perpendicular to $\overline{AD}$. The slope of the fold is $\frac{1}{3}$ because it is perpendicular to $\overline{CC'}$ ($\overline{CC'}$ has a slope of $-3$ using points $C'$ and $C$, and perpendicular lines have slopes negative inverses of each other). Using tangent double angle formula, the slope of $\overline{EC'}$ is $\frac{3}{4}$, which implies $\overline{AE}$ = $\frac{1}{2}$. By the Pythagorean Theorem, $\overline{EC'}=\frac{5}{6}$. It easily follows that our desired perimeter is $\boxed{\textbf{(A)} ~2}$ ~forrestc

Solution 6

It is easy to prove that the ratio of the sum of the larger leg and hypotenuse to the smaller leg depends monotonically on the angle of a right triangle, which means: \[C' F + DF = CF + DF = CD = AD = 3 C'D \implies C' D : DF : C' F = 3 : 4 : 5.\]

For a similar triangle, the ratio of the perimeter to the larger leg is $\frac {3 + 4 +5}{4} = 3.$

$\triangle AEC' \sim \triangle DC'F \implies$ the perimeter of triangle $\bigtriangleup AEC'$ is $3 AC' = \boxed{\textbf{(A)} ~ 2}.$

vladimir.shelomovskii@gmail.com, vvsss

Video Solution by OmegaLearn (Using Pythagoras Theorem and Similar Triangles)

https://youtu.be/cagzLmdbqYQ

~ pi_is_3.14

Video Solution by Interstigation

https://youtu.be/0sEQOjLG-V4

~Interstigation

Video Solution by The Power of Logic

https://www.youtube.com/watch?v=5kbQHcx1FfE

~The Power of Logic

See Also

2021 AMC 10B (Problems • Answer Key • Resources)
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Problem 22
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